The Manorial Documents Register on The National Archives gives these records for Bainton:
https://discovery.nationalarchives.gov.uk/details/c/F237833Many of these at East Riding Archives and Local Studies. But the ones there don't seem to start until
1681.
There is a 1608 survey (rentals) at The National Archives.
And all those at East Riding up to 1753 are indeed part of the Wrangham collection.
It does sound as if the Wrangham family were the local landowners, and if your ancestors were yeomen farmers, they may well have leased land from the family.
Have you found wills for your family? They will often mention payments due for rent, and sometimes mention the land holder's name.
I'm not sure if the East Riding office is open at the moment, but it might be a idea to check with them whether they hold anything prior to 1600. It may be that the Wrangham family only came into possession of the manor in the 1600s, and you would need to find out who owned it before then. As it is not specifically mentioned on The National Archives Manorial Documents Register prior to the 1600s, perhaps it was part of another manor up until them.
There is another page on The NAtional Archives relating to documents at East Ridng about Wrangham's documents at Bainton:
https://discovery.nationalarchives.gov.uk/details/r/b1dcd5f9-6e17-4b89-8f9f-bc2b13ced81cthis again only seems to start in 1597.
It would be well worth you looking at all the linked pages from that page. There looks to be many of them.
If you put in "William Hardy" AND Bainton in the search box on The National Archives, you get this:
https://discovery.nationalarchives.gov.uk/details/r/37658a0c-f10e-4fe0-b8fd-ea6231558de0Not quite the time frame of the 1500s, but pretty exciting!
Looks like this may have been his wife?
https://discovery.nationalarchives.gov.uk/details/r/233d8d19-6f20-4839-995d-f2de733d7b18Keep digging!